Self-similar but not conformally invariant traces obtained by modified Loewner forces

The two-dimensional Loewner exploration process is generalized to the case where the random force is self-similar with positively correlated increments. We model this random force by a fractional Brownian motion with Hurst exponent $Hgeq frac{1}{2}equiv H_{text{BM}}$, where $H_{text{BM}}$ stands for the one-dimensional Brownian motion. By manipulating the deterministic force, we design a scale-invariant equation describing self-similar traces which lack conformal invariance. The model is investigated in terms of the input diffusivity parameter $kappa$, which coincides with the one of the ordinary Schramm-Loewner evolution (SLE) at $H=H_{text{BM}}$. In our numerical investigation, we focus on the scaling properties of the traces generated for $kappa=2,3$, $kappa=4$ and $kappa=6,8$ as the representatives, respectively, of the dilute phase, the transition point and the dense phase of the ordinary SLE. The resulting traces are shown to be scale-invariant. Using two equivalent schemes, we extract the fractal dimension, $D_f(H)$, of the traces which decrease monotonically with increasing $H$, reaching $D_f=1$ at $H=1$ for all $kappa$ values. The left passage probability (LPP) test demonstrates that, for $H$ values not far from the uncorrelated case (small $epsilon_Hequiv frac{H-H_{text{BM}}}{H_{text{BM}}}$) the prediction of the ordinary SLE is applicable with an effective diffusivity parameter $kappa_{text{eff}}$. Not surprisingly, the $kappa_{text{eff}}$s do not fulfill the prediction of SLE for the relation between $D_f(H)$ and the diffusivity parameter.